I was solving some routine problems about determining the possible Jordan forms of a linear operator, given the characteristic and minimal polynomials, and an interesting thought came to my mind! All the combinatorics enthusiasts out there should take a look.
Is there a way to comment on the number of Jordan forms, given the characteristic polynomial of $T:V\to V$?
Let's say $$p_T(t) = \prod_{i=1}^k(t-\lambda_i)^{n_i}$$
is the characteristic polynomial of $T:V\to V$. Is there a closed-form solution to describe the number of Jordan Forms corresponding to this polynomial? Two Jordan forms are considered the same iff they consist of the same Jordan blocks (any permutation).
What if I am also given the minimal polynomial of $T$, namely $m_T(t)?$ $$m_T(t) = \prod_{i=1}^k(t-\lambda_i)^{m_i}$$ where $1\leq m_i\leq n_i$ for all $i=1,2,...,k$
The answer definitely reduces since we have imposed more constraints but by how much? What is the number, exactly?
I think the following ideas will be very important in determining the answer, though I wasn't able to quite figure out something concrete using them:
- The sum of sizes of all Jordan blocks corresponding to $\lambda$ is equal to the multiplicity of $\lambda$ in $p_T(t)$.
- The size of the largest Jordan block corresponding to $\lambda$ is equal to the multiplicity of $\lambda$ in $m_T(t)$.
Thank you, and I look forward to an interesting discussion!
There is not really much more that can be said than what you have observed at the end. A multiset of positive integers whose sum is $n$ is called a partition of $n$, and the number of such partitions is commonly written $p(n)$. So, a Jordan normal form with characteristic polynomial $\prod_{i=1}^k(t-\lambda_i)^{n_i}$ just consists of a partition of $n_i$ for each $i$, so the number of them is $$\prod_{i=1}^kp(n_i).$$ However, there is no known closed form for $p(n)$ (and in the case $k=1$, your problem is equivalent to finding a closed form for $p(n)$).
Similarly, the number of partitions of $n$ into parts such that the largest part is $m$ can be written as $p_m(n)$, so if you additionally require the minimal polynomial to be $\prod_{i=1}^k(t-\lambda_i)^{m_i}$ then the number of such Jordan normal forms is $$\prod_{i=1}^kp_{m_i}(n_i).$$ Again, though, there is no known closed form for $p_m(n)$.